Problem Solving

What does x^2>x translate to ?

My assumption - x^2-x > 0 .So x(x-1)>0.Hence x> 0 and x-1 > 0 or x<0 and x-1<0.But in manhattan , it says the question translated to x < -1 or x > 1 . Can one of you explain ?

You are right it translates to x>1 or x<0

Deepthi, you are right but we need to go a little further to arrive at the final answer.

Let me continue from this : 'x > 0 and x-1 > 0' OR 'x < 0 and x-1 < 0'

x > 0 and x-1 > 0 means we have to find the common solution for x > 0 AND x > 1.
And, The common solution of this is x > 1.

x < 0 and x-1 < 0 means we have to find the common solution for x < 0 and x < 1.
And, the common solution of this is x < 0.

(This can be best understood by making a number line and marking the suitable regions in it)

Therefore, the final solution is x > 1 OR x < 0.

Please also try the following Alternative methods to validate our answer:
1. Substituting numbers
2. The method of Critical Points

it means x cant have values from 0-1
x^2>x
x^2-x>0
x(x-1)>0
x can have only positive values..so make a number line..mark 0 and 1 because they are the zeroes..then u will see that the values x is positive as the values ranging from 1 to infinity and in x-1 values are positive i onward..but not for 1..so 1 can't be the solution.x is positive and x-1 is positive so there product is negative
we will see this in all the 3 regions..
we will see that in the region between 0 and 1 the product will be negative
we will see that in the region between 0 and -infinity the product will be positive
we will see that in the region between 1 and infinity the product will be positive..
HIt thanx if u understand it...

Deepthi Subbu wrote:What does x^2>x translate to ?

My assumption - x^2-x > 0 .So x(x-1)>0.Hence x> 0 and x-1 > 0 or x<0 and x-1<0.But in manhattan , it says the question translated to x < -1 or x > 1 . Can one of you explain ?

x^2 > x only when x is not a proper fraction!
if x is a proper fraction the relation is not true. [1/2]^2 > [1/2]. This is not true.
It not equal to -1 or 1 also.

Press like if you like the above explanation

its true for all fractions except that come between 0 and 1.
x belongs to (-infinity,0)union (1,infinity)..
p.s all real numbers..

Very simple concept, do not follow any specific formula in Inequalities...

x^2>x

x^2 - x > 0 {subtracting x on both sides of the inequality}

x (x - 1) > 0

now understand the meaning of this inequality...

Either both the terms x^2 and x are positive or both the terms are negative as the overall product is a positive. For a product of 2 numbers either both are positive or both are negative...


if both the terms are positive then x has to be greater than 1. then only the overall product will be greater than 0

if both the terms are negative then x has to be less than 0... as x cannot be 0 or 1 in both the cases.

Therefore, it has two possibilities, x>1 or x<0


Hope my post really helped you in understanding the thing...

You are right.